Global representation and multiscale expansion for the Dirichlet problem in a domain with a small hole close to the boundary

Abstract

For each pair of positive parameters, we define a perforated domain by making a small hole of size in an open regular subset of (). The hole is situated at distance from the outer boundary of the domain. Thus, when both the size of the hole and its distance from tend to zero, but the size shrinks faster than the distance. Next, we consider a Dirichlet problem for the Laplace equation in the perforated domain and we denote its solution by Our aim is to represent the map that takes to in terms of real analytic functions of defined in a neighborhood of (0, 0). In contrast with previous results valid only for restrictions of to suitable subsets of we prove a global representation formula that holds on the whole of Such a formula allows us to rigorously justify multiscale expansions, which we subsequently construct.